Reproducibility of the Empty Marking
Kurt Lautenbach
11/2001
Fachberichte
INFORMATIK
Universit ¨at Koblenz-Landau
Institut f ¨ur Informatik, Rheinau 1, D-56075 Koblenz
E-mail: [email protected],Reproducibility of the Empty Marking
Kurt LautenbachUniversity of Koblenz-Landau, Universit¨atsstr. 1, 56075 Koblenz, Germany, [email protected],
WWW home page:http://www.uni-koblenz.de/~laut
Abstract. The main theorem of the paper states that the empty mark-ing∅is reproducible in a p/t-netN if and only if there are reproducing T-invariants whose net representations have neither traps nor co-traps (deadlocks, siphons). This result is to be seen in connection to modeling processes, since all processes which have a start and a goal event usually reproduce the empty marking.
1
Introduction
This paper deals with a necessary and sufficient condition for the reproducibility of the empty marking in place/transition nets. First of all, this is a theoretical result. It will be shown that there are interesting connections to other notions, for example to the forward and backward liveness of the empty marking. On the other hand, there are application oriented aspects of that result. All processes with a start and a goal event usually reproduce the empty marking. Examples can be found in flexible manufacturing, workflow modeling, automatic control, ets. Nevertheless, to our knowledge there are no specific results dealing with the reproducibility of the empty marking.
The paper is organized as follows. In the second section one finds the net theo-retic preliminaries. The main theorem of the third section provides a necessary and sufficient condition for the reproducibility of the empty marking∅.
I am greatly indebted to J¨org M¨uller, Lutz Priese, Carlo Simon, Einar Smith and Paula Swatman for valuable comments and for pushing me to publish the material.
2
Net Theoretical Preliminaries
In this section some fundamentals of net theory will be (very) briefly introduced.
2.1 Place/Transition Nets
Definition 1. 1. Aplace/transition net (p/t-net)is a quadrupleN = (S, T, F, W)
(a) S andT are finite, non empty, and disjoint sets.S is the set of places
(in the figures represented by cycles).T is the set of transitions (in the figures represented by boxes).
(b) F ⊆(S×T)∪(T×S)is the set of directed arcs. (c) W :F→IN\{0}assigns aweightto every arc.
In case ofW :F → {1}, we will writeN = (S, T, F)as an abridgement. 2. The preset (postset) of a nodex∈S∪T is defined as•x={y∈S∪T|(y, x)∈
F} (x•={y∈S∪T|(x, y)∈F}).
The preset (postset) of a setH ⊆S∪T is•H=Sx∈H •x H•=Sx∈Hx•. For all x∈S∪T it is assumed that |•x|+|x•| = 1 holds; i.e. there are
no isolated nodes.
3. A placep(transitiont) issharediff|•p|=2 or|p•|=2 (|•t|=2 or |t•|=2). 4. A place pis an input (output) boundary placeiff•p=∅(p•=∅).
5. A transitiont is aninput (output) boundary transition iff•t=∅(t•=∅). Definition 2. Let N = (S, T, F, W)be ap/t-net.
1. Amarkingof N is a mapping M :S−→IN. M(p)indicates the number of
tokensonpunderM.p∈S ismarked byM iffM(p)=1.H ⊆Sismarked byM iff at least one place p∈H is marked by M. Otherwise p andH are
unmarked, respectively.
2. A transitiont∈T isenabledby M, in symbols M[ti, iff
∀p∈•t:M(p)=W((p, t)).
3. If M[ti, the transition t may fire or occur, thus leading to a new marking
M0, in symbols M[tiM0, with M0(p) := M(p)−W((p, t)) if p∈•t\t• M(p) +W((t, p)) if p∈t•\•t M(p)−W((p, t)) + W((t, p)) if p∈•t∩t• M(p) otherwise for allp∈S.
4. The set of all markings reachable from a marking M0, in symbols [M0i, is the smallest set such that
M0∈[M0i
M∈[M0i ∧ M[tiM0 =⇒M0∈[M0i.
[M0iis also called the set of follower markingsofM0.
5. σ = t1. . . tn is a firing sequence or occurrence sequence for transitions
t1, . . . , tn∈T iff there exist markingsM0, M1, . . . , Mn such that M0[t1iM1[t2i. . .[tniMnholds;
in short M0[σiMn.M0[σi denotes thatσ starts from M0. The firing count σ(t)oftinσindicates how oftentoccurs inσ. The (column) vector of firing counts is denoted byσ.
6. The pair(N, M0)for some marking M0 of N is ap/t-systemor a marked p/t-net.M0 is the initial marking.
7. A markingM∈[M0iisreproducibleiff there exists a markingM0∈[Mi, M0 6= M s.t. M∈[M0i.
Definition 3. Let N = (S, T, F, W)be ap/t-net andM0 a marking of N. 1. A transition t∈T is live under M0 or in (N, M0) iff ∀M ∈[M0i ∃M0∈
[Mi:M0[ti.
2. A transitiont isdeadin(N, M0) iff ∀M∈[M0i:t is not enabled.
(N, M0)orM0 isdead iff 6 ∃t∈T :M0[ti.
3. (N, M0)orM0 isweakly live(deadlock-free) iff ∀M∈[M0i ∃t∈T :M[ti. 4. (N, M0)orM0 islive iff ∀t∈T : tis live underM0.
5. A place p∈S isbounded underM0 iff ∃k∈IN ∀M∈[M0i:M(p)5k.
(N, M0)orM0 isbounded iff ∀p∈S:pis bounded underM0. 6. A place pismarkablein(N, M0) iff ∃M∈[M0i:M(p)>0.
A setA⊆S ismarkablein(N, M0) iff ∃p∈A:pis markable in(N, M0).
2.2 Place Vectors and Transition Vectors Definition 4. Let N = (S, T, F, W)be ap/t-net;
1. N ispure iff 6 ∃(x, y)∈(S×T)∪(T×S) : (x, y)∈F∧(y, x)∈F. 2. Aplace vector (|S|-vector)is a column vector v:S−→Zindexed by S. 3. A transition vector (|T|-vector) is a column vector ω :T −→ Z indexed by
T.
4. Theincidence matrixof N is a matrix [N] :S×T −→Z indexed byS and
T such that [N](p, t) = −W((p, t)) if p∈•t\t• W((t, p)) if p∈t•\•t −W((p, t)) + W((t, p)) if p∈•t∩t• 0 otherwise
Column vectors of which all entries equal 0 (1) are denoted by0 (1).vt andAt are the transposes of a vector v and a matrix A, respectively. The columns of
[N]are |S|-vectors, the rows of [N]are transposes of |T|-vectors. Markings are representable as|S|-vectors, firing count vectors as|T|-vectors. The|S|-vector0 denotes the empty marking∅.
Remark 1. In the following, we assume that all p/t-nets are pure. The reason for that is that we want a one-to-one correspondence betweenp/t-nets and their incidence matrices. This is no major restriction since
p k l can be replaced by p k l
without altering relevant behaviour characteristics.
Definition 5. Letibe a place vector andja transition vector ofN = (S, T, F, W). 1. iis aplace invariant (p-invariant) iff i6=0andit·[N] =0t
2. j is atransition invariant (t-invariant) iff j6=0and[N]·j=0
3. ||i|| ={p∈S|i(p)6= 0} and ||j|| = {t∈T|j(t)6= 0} are the supports of i andj, respectively.
4. Ap-invarianti(t-invariantj) is
• non-negative iff ∀p∈S:i(p)=0 (∀t∈T :j(t)=0) • positive iff ∀p∈S:i(p)>0 (∀t∈T :j(t)>0) • minimal iff i(j)is non-negative
and6 ∃p-invarianti0:||i0||&||i|| (6 ∃t -invariantj0 :||j0||&||j||)
and the greatest common divisor of all entries ofi(j)is1
5. Thenet representationNi= (Si, Ti, Fi, Wi) of ap-invariantiis defined by
Si:=||i|| Ti:=•Si∪Si•
Fi:=F ∩((Si×Ti)∪(Ti×Si)) Wi is the restriction of W to Fi.
Thenet representationNj = (Sj, Tj, Fj, Wj) of at-invariantj is defined by
Tj :=||j|| Sj :=•Tj∪Tj•
Fj :=F∩((Sj×Tj)∪(Tj×Sj)) Wj is the restriction of W to Fj.
6. N iscoveredby a p-invarianti (t-invariantj) iff ∀p∈S:i(p)6= 0 (∀t∈ T:j(t)6= 0)
Proposition 1. Let (N, M0)be ap/t-system,iap-invariant; then
Proposition 2. Let (N, M0) be a p/t-system, M1∈[M0i a follower marking of M0, and σ a firing sequence that reproduces M1: M1[σiM1; then the firing count vector σof σis at-invariant.
Definition 6. Let N = (S, T, F, W) be a p/t-net, M0 a marking of N, and
r=0a|T|-vector;r isrealizable in (N, M0)iff there exists a firing sequenceσ with M0[σiandσ=r.
Proposition 3. LetN = (S, T, F, W)be ap/t-net,M1 andM2 markings ofN, andσ a firing sequence s.t.M1[σiM2; then the linear relation
M1+ [N]σ=M2 holds.
In the above linear relation, thestate equation, the order of transition firings is lost.
LetH be a set of places (transitions); the|S|-vector (|T|-vector)χ(H) with χ(H)(h) =
1 if h∈H 0 if h6∈H is thecharacteristic vectorof H.
A set of placesH ismarkable in (N, M0) iff there is a markingM∈[M0i s.t.χ(H)t· M >0.
Next, we will introduce the reachability graph or marking graph. Because of their size, reachability graphs are unsuitable for practical problems. As a conceptual means, they are indispensable.
Definition 7. Let N = (S, T, F, W)be ap/t-net andM0 a marking of N; the reachability graph RG(N, M0) = (V, E) of (N, M0)is a directed graph with
V = [M0i as set of nodesand
E=
n
(M, t, M0)|M∈[M0i ∧ t∈T ∧M[tiM0
o
as set of labeled arcs. L(E) =nt∈T| ∃(M, t, M0)∈E
o
is the set of labels.
2.3 Traps, Co-Traps and Liveness
Definition 8. Let N = (S, T, F, W)be ap/t-net;
1. A non-empty setH ⊆S of places is atrap iff H•⊆ •H.
2. A non-empty set H ⊆S of places is aco-trap(or a deadlock, siphon) iff
3. A trap (co-trap) h isminimal iff there is no trap (co-trap) contained in
H as a proper subset.
Once a trap is marked it remains marked. Once a co-trap is unmarked it remains unmarked. The supports||i||of all non-negativep-invariantsiare traps and co-traps.
Co-traps play an outstanding role for the weak liveness of markedp/t-nets. This is demonstrated by the following two lemmata.
Definition 9. Let (N, M)be ap/t-system withN = (S, T, F, W); letp∈S and
D⊆S;
pist-undermarked byM for somet∈T iffM(p)< W(p, t);
pis undermarked byM iff there exists a t∈T such that pist-undermarked by M.
D isundermarked byM iff allp∈D are undermarked byM. Lemma 1. Let N = (S, T, F, W)be a p/t-net andM a marking of N; if (N, M)is dead the set of all places that are undermarked by M is a co-trap. Proof. Let (N, M) be dead. If the setH :={p∈S|pundermarked byM}is no co-trap, then there is a transitionu∈•H s.t. u6∈H•. This implies •u∩H =∅ saying that all input places ofuare not undermarked byM. So, all places of•u are for not∈T-undermarked byM, in particular not foruitself. Consequently, uis enabled byM in contrast to the fact that (N, M) is dead.
Lemma 2. Let N = (S, T, F, W)be a p/t-net andM0 a marking ofN; if no minimal co-trap of N can get undermarked, (N, M0)is weakly live. Proof. Suppose, (N, M0) is not weakly live. Then there is a markingM∈[M0i s.t. (N, M) is dead. Then H :={p∈S|pis undermarked by M} is a co-trap. Consequently the minimal sub-co-traps ofH are undermarked byM in contrast to the assumption. So (N, M0) is weakly live.
Clearly, in nets whose arc weights are equal to 1, p∈S and D ⊆ S are undermarked by a markingM iffM(p) = 0 andM(s) = 0 for alls∈D hold.
2.4 Synchronization Graphs
Definition 10. A p/t-net N = (S, T, F) is a synchronization graph (marked graph) iff for allp∈S
|•p|=|p•| = 1 holds.
For synchronization graphs the following well known statements hold.
Lemma 3. Let G= (S, T, F)be a synchronization graph andM0a marking of
G;
1. H⊆S is a trap or a co-trap iff H is the place set of a cycle. 2. (G, M0)is live iff all cycles are marked byM0.
3
Reproduction
Subject of this section is the following reproduction theorem and its proof. The theorem provides a necessary and sufficient condition for the reproducibility of the empty marking in ap/t-netN = (S, T, F).
Theorem 1 (Reproduction Theorem). LetN = (S, T, F)be ap/t-net and
r a non-negativet-invariant ofN;
then inN the empty marking∅is reproducible by realizing k·rfor k∈IN\{0}
iff the net representation Nrof r does neither contain a trap nor a co-trap.
We start proving by showing that the reproduction condition, i.e. the non-existence of a trap or a co-trap inNr, is necessary. For that we need the following lemma 4 and theorem 2 whose corollary 3 states the necessity.
Lemma 4. LetN = (S, T, F)be ap/t-net and∅(the empty marking) its initial marking;
then every transition that is firable at all in(N,∅)is firable arbitrarily often in
(N,∅).
Proof. Lett∈T be a firable transition of (N,∅). Then a firing sequenceσexists with t as last transition and such that ∅[σiholds. Then M[σiholds for every other markingM ofN such thatσ can be repeated arbitrarily often.
Theorem 2. Let N = (S, T, F)be a p/t-net;
(N,∅)is live iff N contains no co-trap.
Proof. Let (N,∅) be live. Then all transitions are firable. So, there are no places p∈S with •p=∅(input boundary places). Consequently and since there exist no isolated nodes, every place is output place of some transition and can thus be marked. This, however, is impossible ifN contains a co-trapD. In that case D is empty under∅ and none of its places can be marked.
Assume now thatN contains no co-trap. Then there are no input boundary places because every such place is a minimal co-trap. Let U be the set of un-markable places of (N,∅) and let U 6= ∅. Then •U 6= ∅ sinceU contains no input boundary places. Let bet∈•U. Thentis not firable. If all input places of tcan be marked at all, they can be marked sufficiently high according to lemma 4. So, the reason for the non-firability of t must be an input place pof t that cannot be marked in (N,∅). Consequently,p∈U, t∈U•, and •U ⊆ U• hold such thatU turns out to be a co-trap in contrast to the assumption. So,U must be empty. Then, all places are markable sufficiently high such that all transitions are firable arbitrarily often, i.e. (N,∅) is live.
Corollary 1. LetN = (S, T, F)be ap/t-net;
(The “backward liveness” of (N, M) is the liveness of (N0, M) whereN0 arises formN by reversing the direction of all arcs.)
Proof. That is an immediate consequence of the fact that traps (co-traps) are transformed into co-traps (traps) if the direction of all arcs is changed.
This corollary guarantees for example that the reproduction condition (a t-invariantr = ∅exists whose net representationNrneither contains a trap nor a co-trap) is necessary and sufficient for the liveness of (Nr,∅) in both directions.
Corollary 2. LetN = (S, T, F)be ap/t-net;
there exists a dead transition in(N,∅)iffN contains a co-trap.
Proof. If (N,∅) contains a dead transition, (N,∅) is not live and thus contains a co-trap according to theorem 2.
IfN contains a co-trapD,Dis empty under∅such that all transitionst∈D• are dead in (N,∅).
Corollary 3. Let N = (S, T, F) be a p/t-net and r = 0 a t-invariant of N such that the empty marking∅can be reproduced by realization ofk ·rfor some
k∈IN\{0};
then the net representation Nr ofr neither contains a trap nor a co-trap. Proof. If ∅ can be reproduced by realization of k·r for k∈IN\{0},(Nr,∅) is live in both directions. ThenNr contains no trap and no co-trap according to corollary 1.
This completes the necessity part of the proof of theorem 1.
It is more intricate to show that the reproduction condition of theorem 1 is also sufficient for the reproducibility of ∅. The basic idea for that part of the proof is an induction on the number of shared places. We therefore need some details aboutp/t-nets without shared places.
Lemma 5. Let N = (S, T, F)be a p/t-net without shared places;
then N contains neither a trap nor a co-trap iffN is a synchronization graph without cycles.
Proof. Since N does not contain shared places|•p|51 and |p•|51 holds for all p∈S. If N does not contain traps and co-traps there exist no boundary places because if p is a boundary place {p} is either a co-trap or a trap. So, |•p|=|p•|= 1 holds for allp∈S, i.e.N is a synchronization graph. According to lemma 3N does not contain cycles.
IfN is a synchronization graph without cyclesN consists of chains beginning and ending with boundary transitions. Assume there is a co-trapDinN andp∈ D. Sincepis no input boundary place there is a chainK={k0, k1, k2, . . . , k2l, p} such thatk1, k3, . . . , k2l−1, p are places ofD andk0 is an input boundary tran-sition. But thenk0∈•D andk0 6∈ D• in contrast to the assumption thatD is a co-trap.
The next theorem and its corollary confirm that forp/t-nets without shared places the reproduction condition of theorem 1 is sufficient for reproducing the empty marking∅.
Theorem 3. Let G= (S, T, F) be a synchronization graph without cycles and
M a marking ofG;
then every integer |T|-vectorv=0is realizable in(G, M)iff
M + [G] · v = 0.
Proof. Let v = 0 be realizable in (G, M), i.e. there exists a firing sequence σ withM[σi(M+ [G]·v) s.t. all transitionstoccur inσas often asv(t) indicates. As a markingM + [G]·vis non-negative.
Let now beM+[G]·v=0. v = 0is trivially realizable by firing no transition at all.
In casev 0a transitionz0∈Texists withv(z0)> 0. Ifz0is not enabled by M and thus not suited to be the first transition of a realization ofv there exists a placep0∈•z0withM(p0) = 0. InM+ [G]·v=0one inequality belongs top0, namelyM(p0)−v(z0)+v(z1)=0 with{z1} = •p0. This impliesv(z1)=v(z0)> 0. If z1 is not enabled for M, too, there is a transitionz2 with v(z2) >0 etc. By continuing thus one gets a backwards directed chain of transitionsz0, . . . , zn, wherev(zi)> 0,05 i5 n. The chain is finite becauseGis finite and cycle-free. The fact thatzn is enabled forM is ensured because eventually the chain ends in an input boundary transition that is enabled trivially.
Lety1∈T be a transition withv(y1)> 0 that is enabled forM. Let further-moreχ(y1) be the characteristic|T|-vector ofy1. Then
M+ [G]·v=M+ [G]·χ(y1) + [G]·(v−χ(y1)) =M+G(y1) + [G]·(v−χ(y1))=0
where G(y1) is the column of G belonging to y1. M1 := M +G(y1) is the follower marking ofM after one firing ofy1. Then v1 :=v−χ(y1) is that part ofv that still must be realized. For k := P
t∈T v(t) the following holds
X
t∈T
v1(t) = k−1.
In casek−1>0, the procedure has to be repeated forM1andv1. . .. After ksteps the algorithm ends with vk = 0.
The sequenceσ=y1. . . yk that transformsM intoMk is a realization ofv. Every transitiony occurs inσ v(y) times.
Corollary 4. Let G= (S, T, F) be a synchronization graph without cycles and letr0be a |T|-vector;
then in G the empty marking ∅ can be reproduced by realizing r iff r is a t -invariant of G.
Proof. If∅can be reproduced by realizingr, r is at-invariant.
Ifr0is at-invariant0+ [G]·r=0holds. Due to theorem 3ris realizable in (G,∅); it is a reproduction of∅ inG.
A well known result ([Murata89], theorem 16) says that in an acyclic Petri net N B is reachable from A iff there exists a non-negativ integer solutionx satisfying
B=A+ [N] · x
This means that for synchronization graphs without cycles the existence of a t-invariantr 0is necessary and sufficient for the reproducibility of the empty marking. So far we get a proof of theorem 1 forp/t-nets without shared places. To start off handlingp/t-nets with shared places we will concentrate ont-invariants because those parts of the nets which are not belonging to at-invariant cannot contribute to a reproduction of markings. So, we will assume for some time that the nets are covered by a t-invariantr >0. Moreover, we assumer=1. This is no further restriction because in casesr(t) = kt = 2 kt−1 copies oftcan be added to the original net thus getting a new net with equal behaviour in which
1is a t-invariant. For example, in the following netN r= (1,1,1,2,1,1,1)tis a t-invariant. Adding one copy of transitiont4 results in the netN0 where 1is the t-invariant corresponding tor.
p1 p2 p 3 p 4 p 5 p 6 p 7 t 1 t2 t3 t4 t 5 N t 7 t6 p1 p2 p 3 p 4 p 5 p 6 p 7 t 1 t2 t3 t4 t 5 N 0 t 7 t6 t4 0
In nets with arc weights equal to 1, a further profitable consequence of this assumption is that|•p|=|p•|holds for all placesp∈S, that is every places has as many ingoing as outgoing arcs.
The next theorem shows that the reproduction condition of theorem 1 is suffi-cient if1is a t-invariant.
Theorem 4. Let N = (S, T, F) be a p/t-net that contains neither traps nor co-traps and let 1be a t-invariant of N;
then the empty marking is reproducible in N by realization of k·1 for some
k∈IN\{0}.
Proof. The proof is based on an induction on the number mof shared places. Form= 0 the statement of the theorem follows from corollary 4.
Supposing the statement is true forp/t-nets withm−1 shared places. Letp be a shared place ofN and letN(−p)= (S(−p), T, F(−p)) be the net that results fromN by omittingp. ThusS(−p):=S\{p}, F(−p):=F\(({p}×T)∪(T×{p})). ThenN(−p)is ap/t-net for which the following holds:
(a) N(−p)contains no trap and no co-trap since every trap (co-trap) ofN(−p)is a trap (co-trap) ofN.
(b) The |T|-vector1 is also at-invariant ofN(−p) because [N(−p)] arises from [N] by omitting the row belonging top.
(c) In N(−p) the empty marking is reproducible by realizingk(−p)·1 for some k(−p) ∈IN\{0}sinceN(−p)hasm −1 shared places.
In a realization ofk(−p)·1inN(−p)every transition firesk(−p)times. Adding in N and N(−p) to every transition (k(−p)−1) copies of itself one obtains the netsN∗= (S, T∗, F∗) andN(∗−p)= (S(−p), T∗, F(∗−p)), respectively.
InN∗ the|T∗|-vector1is at-invariant because [N∗] = [N]. . .[N]
| {z }
k(−p) times
. Fur-thermore, N∗ does not contain traps and co-traps as N does not; (N∗,∅) is therefore live in both directions (corollary 1).
InN(∗−p) the empty marking is reproducible by realizing the |T∗|-vector 1
(because of (c)), i.e. everyt∈T∗fires exactly once.
Letσ∗ be a firing sequence in N(∗−p)that realizes the|T∗|-vector1, thus re-producing the empty marking. Of course,σ∗ is not necessarily a firing sequence in (N∗,∅) because there might be markings in whichpis not sufficiently mark-able to enmark-able transitions of p•. So pmight be kind of a bottleneck in (N∗,∅). Nevertheless, we will exploit σ∗ to some extent for (N∗,∅). That is, we will transform σ∗ into another firing sequenceσ0 by reversing the order of concur-rent subsequences such thatσ0 is also a realization of1inN(∗−p)that reproduces the empty marking.
Whetherσ∗is a firing sequence in (N∗,∅) or not depends on the existence of cycles throughpin N∗. Clearly, if there don’t exist cycles throughpin N∗ σ∗ is a firing sequence that reproduces the empty marking in N∗.
Let
T•(p) :={t∈T∗|(∃n = 0 : (t, p)∈F2n+1)∧
(∃p0 ∈t• : (∀t0∈(p0)• : t0 ∈T•(p))∨ p0 = p)}
be the set of transitions whose firing causes necessarily a flow of tokens through p.
Let
•T(p) :={t∈T∗|(∃n =0 : (p, t)∈F2n+1)∧
be the set of transitions which cannot be activated without tokens flowing throughp.
Then
C(p) := T•(p) ∩•T(p)
is the set of transitions which necessarily transport tokens on cycles throughp. Next, we partition the transitions of•p∪p•w.r.t. their membership ofC(p) :
D(p) :=•p∩ C(p), D0(p) := p• ∩ C(p) I(p) :=•p\C(p), I0(p) := p•\C(p)
Since (N∗,∅) is live it is possible to lead up at least one token to pwithout firing a transition ofC(p). Consequently I(p) 6= ∅. Let σ1 be a firing sequence inN∗with∅[σ1iwhere as many as possible transitions fire exactly once without any firing of a transition ofC(p).σ1 which is also a firing sequence of (N(∗−p),∅) is the first part of σ0. In other words, σ1 arises form a subsequence of σ∗ by rearranging concurrently enabled transitions.
Similarly, since (N∗,∅) is also backwards live it is possible to lead up at least one token topin backward direction without firing a transition ofC(p). So I0(p)6=∅. Let%be the backward firing sequence in (N∗,∅) starting at∅where as many transitions as possible fire exactly once without any firing of a transition of C(p). Then%in reverse direction less all transitions already occurring in σ1 is the third (and last) part σ3 of σ0. The middle part σ2 of σ0 is σ∗ less all transitions already occurring in σ1 and σ3. σ0 =σ1σ2σ3 is a firing sequence of (N(∗−p),∅) but not necessarily of (N∗,∅) because σ1 might not pump enough tokens intopto makeσ2σ3possible.
We now aim at finding ak0∈IN\{0} such that k0σ0 := (k0σ 1)(k0σ2)(k0σ3) :=σ| {z }1. . . σ1 k0 σ2. . . σ2 | {z } k0 σ3. . . σ3 | {z } k0
is a realization ofk0·1in (N∗,∅) thus reproducing the empty marking.
Since1is at-invariant ofN∗ |•p|=|p•|holds. Moreover, because ofI(p) ∪ D(p) = •pandI0(p)∪ D0(p) = p•the following holds
k0· |•p|=k0(|I(p)|+|D(p)|) =k0(|I0(p)|+|D0(p)|) =k0|p•| for allk0∈IN and
−|D0(p)|+|D(p)|=|I0(p)| − |I(p)|. We distinguish two cases:
First case:−|D0(p)|+|D(p)|=0,
that is the token count on p caused by (k0σ1) is not decreased by (k0σ2). In order to get sufficiently many tokens onpby (k0σ1) to make (k0σ2) possible the restriction onk0 isk0|I(p)|=|D0(p)|and
k0 =|D0(p)| |I(p)| .
Such ak0 exists because ofI(p)6=∅. Second case:−|D0(p)|+|D(p)|<0,
i.e. the token count onpis decreased during (k0σ2). Then, after (k0σ1)((k0−1)σ2) sufficiently many tokens have to be onp to makeσ2 once more possible. That means for the token count onp
|D0(p)|5k0|I(p)| + (k0−1)(−|D0(p)| + |D(p)|) =k0|I(p)| + (k0−1)(|I0(p)| − |I(p)|) =|I(p)| + (k0−1)|I0(p)|
k0= |D0(p)| − |I(p)| |I0(p)| + 1
Such ak0 exists because ofI0(p)6=∅. Altogether, k0 =max 1,|D0(p)| |I(p)| in first case k0 =max 1,D0(p)| − |I(p)| |I0(p)| + 1 in second case
For the original p/t-net N the empty marking is reproducible by realization k(−p)·k0·1.
Example 1. In the net of figure 1 the following holds:
T•(p 1) =T\ {tB} •T(p 1) =T\ {tA} C(p1) =T•(p1)∩ •T(p1) = T\{tA, tB} D(p1) =•p1 ∩ C(p1) = {t1, t2} D0(p 1) =p•1 ∩ C(p1) = {t4, t5} I(p1) =•p1\C(p1) = {tA} I0(p 1) =p•1\C(p1) = {tB}
p 2 p 1 p 3 p 4 p 5 t 1 t A t 2 t 3 t 4 t 5 t B 6 Figure 1 So we find |I(p1)|=|I0(p1)| = 1 |D(p1)|=|D0(p1)| = 2 −|D0(p 1)|+|D(p1)|= 0 (first case) k= max 1, 2 1 = 2 is the minimal possible multiple.
(2σ) = (2σ1)(2σ2)(2σ3)
= (tAtA)(t5t4t3t2t1 t5t4t3t2t1) (tBtB) reproduces the empty marking.
Example 2. In the net of figure 2 the following holds T•(p 1) =T\{tB} •T(p 1) =T\{tA, t1} C(p1) =T•(p1)∩ •T(p1) = T\{tA, tB, t1} D(p1) =•p1 ∩C(p1) = {t2} D0(p 1) =p•1 ∩ C(p1) = {t4, t5} I(p1) =•p1\C(p1) = {tA, t1} I0(p 1) =p•1\C(p1) = {tB}
p 2 p 1 p 3 p 4 p 5 1 A 2 t 3 t 4 t 5 t B Figure 2 So we find |I(p1)|=|D0(p1)| = 2 |I0(p 1)|=|D(p1)| = 1 −|D0(p 1)|+|D(p1)|=−1 (second case) k= max 1, 2−2 1 + 1 = 1 is the minimal possible multiple.
σ= (σ1)(σ2)(σ3)
= (tAt1)(t5t4t3t2) (tB) reproduces the empty marking.
Proof (of Theorem 1). Letr ∅ be at-invariant and letNr = (Sr, Tr, Fr) be the corresponding net representation that does not contain a trap or a co-trap. Then the netN∗= (Sr, Tr∗, Fr∗), that arises fromNrby adding (r(t)−1) copies of every transition t∈Tr, does also not contain a co-trap or a trap. Moreover, the|T∗|-vector1 is at-invariant ofNr∗.
Due to theorem 4 inNr∗the empty marking is reproducible by realization of k · 1 for somek∈IN\{0}. Then inNr the empty marking is reproducible by a corresponding realization ofk·r, i.e. whenever inNr∗ a copy oft∈Tr fires in Nr t fires itself. So, the reproduction condition is sufficient. Its necessity is an immediate consequence of corollary 3.
The section will be concluded by two corollaries.
Corollary 5. LetN = (S, T, F) be ap/t-net, r∅ at-invariant, and Nr its net representation;
then the empty marking is reproducible by realization of k · r for some k∈
IN\{0}iff Nr is live in both directions.
Proof. Immediate consequence of theorem 4 and corollary 1.
The next corollary extends the result of this section top/t-nets of the form N = (S, T, F, W).
Corollary 6. Let N = (S, T, F, W) be a p/t-net and r ∅ a t-invariant of
N;
then in N the empty marking is reproducible by realizing k · r for some k∈
IN\{0}iff the net representationNr ofrneither contains a trap nor a co-trap.
Proof. If in N the empty marking is reproducible by realization of k · r Nr neither contains a trap nor a co-trap.
IfNrneither contains a trap nor a co-trap the following transformation ofNr into ap/t-netNr0 whose arc labels all are equal to 1 does not alter that property.
t n p t 1 t 2 t n p ::: ::: ::: ::: : : : istransformedinto
t n p t 1 t 2 t n p ::: ::: : : : istransformedinto
InNr0 a t-invariantr0 exists as a counterpart to r. Since in Nr0 the empty marking can be reproduced by realizingk · r0 this is possible inNrby realizing k · r.
4
Conclusion
In this paper a necessary and sufficient condition for the reproducibility of the empty marking was introduced. In essence, the condition states that the net representations of the relevant T-invariants should have neither traps nor co-traps.
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